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Mirrors > Home > MPE Home > Th. List > al0ssb | Structured version Visualization version GIF version |
Description: The empty set is the unique class which is a subclass of any set. (Contributed by AV, 24-Aug-2022.) |
Ref | Expression |
---|---|
al0ssb | ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | 0ex 5308 | . . 3 ⊢ ∅ ∈ V | |
2 | sseq2 4009 | . . . 4 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 ⊆ ∅)) | |
3 | ss0b 4398 | . . . 4 ⊢ (𝑋 ⊆ ∅ ↔ 𝑋 = ∅) | |
4 | 2, 3 | bitrdi 287 | . . 3 ⊢ (𝑦 = ∅ → (𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅)) |
5 | 1, 4 | spcv 3596 | . 2 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 → 𝑋 = ∅) |
6 | 0ss 4397 | . . . 4 ⊢ ∅ ⊆ 𝑦 | |
7 | 6 | ax-gen 1798 | . . 3 ⊢ ∀𝑦∅ ⊆ 𝑦 |
8 | sseq1 4008 | . . . 4 ⊢ (𝑋 = ∅ → (𝑋 ⊆ 𝑦 ↔ ∅ ⊆ 𝑦)) | |
9 | 8 | albidv 1924 | . . 3 ⊢ (𝑋 = ∅ → (∀𝑦 𝑋 ⊆ 𝑦 ↔ ∀𝑦∅ ⊆ 𝑦)) |
10 | 7, 9 | mpbiri 258 | . 2 ⊢ (𝑋 = ∅ → ∀𝑦 𝑋 ⊆ 𝑦) |
11 | 5, 10 | impbii 208 | 1 ⊢ (∀𝑦 𝑋 ⊆ 𝑦 ↔ 𝑋 = ∅) |
Colors of variables: wff setvar class |
Syntax hints: ↔ wb 205 ∀wal 1540 = wceq 1542 ⊆ wss 3949 ∅c0 4323 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1798 ax-4 1812 ax-5 1914 ax-6 1972 ax-7 2012 ax-8 2109 ax-9 2117 ax-ext 2704 ax-nul 5307 |
This theorem depends on definitions: df-bi 206 df-an 398 df-tru 1545 df-fal 1555 df-ex 1783 df-sb 2069 df-clab 2711 df-cleq 2725 df-clel 2811 df-v 3477 df-dif 3952 df-in 3956 df-ss 3966 df-nul 4324 |
This theorem is referenced by: iota0def 45748 aiota0def 45804 |
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